Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 99.9%

Time: 5.1s

Alternatives: 11

Speedup: 6.7×

Specification

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x\_m - -1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.009:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -1\right) \cdot 0.5\right)}^{2}}{1 + t\_0}}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (cos (atan x_m)) -1.0) 0.5)))
   (if (<= x_m 0.009)
     (*
      (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
      (* x_m x_m))
     (/
      (/
       (- 1.0 (pow (* (- (sqrt (/ 1.0 (fma x_m x_m 1.0))) -1.0) 0.5) 2.0))
       (+ 1.0 t_0))
      (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (cos(atan(x_m)) - -1.0) * 0.5;
	double tmp;
	if (x_m <= 0.009) {
		tmp = fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = ((1.0 - pow(((sqrt((1.0 / fma(x_m, x_m, 1.0))) - -1.0) * 0.5), 2.0)) / (1.0 + t_0)) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(cos(atan(x_m)) - -1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.009)
		tmp = Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(Float64(1.0 - (Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) - -1.0) * 0.5) ^ 2.0)) / Float64(1.0 + t_0)) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.009], N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00899999999999999932

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2} + \frac{1}{8}\right) \cdot {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      13. lower-*.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.00899999999999999932 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{2} \cdot \left(\cos \tan^{-1} x + 1\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      2. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right)}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \color{blue}{\left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      4. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \color{blue}{\cos \tan^{-1} x}\right)}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      5. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \left(\frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}{1 + \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \left(\frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right) \cdot \left(\frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}{1 + \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\left(\cos \tan^{-1} x - -1\right) \cdot 0.5\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot 0.5}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
   7. Step-by-step derivation

      1. cos-atan-rev N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{1 + x \cdot x}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      3. sqrt-undiv N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\color{blue}{\sqrt{\frac{1}{1 + x \cdot x}}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\color{blue}{\sqrt{\frac{1}{1 + x \cdot x}}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\sqrt{\color{blue}{\frac{1}{1 + x \cdot x}}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{1 + \color{blue}{{x}^{2}}}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{\color{blue}{{x}^{2} + 1}}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{\color{blue}{x \cdot x} + 1}} - -1\right) \cdot \frac{1}{2}\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot \frac{1}{2}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      9. lower-fma.f64 99.9

         \[\leadsto \frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} - -1\right) \cdot 0.5\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot 0.5}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{1 - {\left(\left(\color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}} - -1\right) \cdot 0.5\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot 0.5}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.009:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}} - -1\right) \cdot 0.5\right)}^{2}}{1 + \left(\cos \tan^{-1} x - -1\right) \cdot 0.5}}{1 + \sqrt{\left(\cos \tan^{-1} x - -1\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m - -1\\ \mathbf{if}\;x\_m \leq 0.0112:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0 \cdot 0.5}{\mathsf{fma}\left(\sqrt{t\_0}, \sqrt{0.5}, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (cos (atan x_m)) -1.0)))
   (if (<= x_m 0.0112)
     (*
      (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
      (* x_m x_m))
     (/ (- 1.0 (* t_0 0.5)) (fma (sqrt t_0) (sqrt 0.5) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m)) - -1.0;
	double tmp;
	if (x_m <= 0.0112) {
		tmp = fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - (t_0 * 0.5)) / fma(sqrt(t_0), sqrt(0.5), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(cos(atan(x_m)) - -1.0)
	tmp = 0.0
	if (x_m <= 0.0112)
		tmp = Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 * 0.5)) / fma(sqrt(t_0), sqrt(0.5), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0112], N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0111999999999999999

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2} + \frac{1}{8}\right) \cdot {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      13. lower-*.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0111999999999999999 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1 - \color{blue}{\frac{1}{2} \cdot \left(\cos \tan^{-1} x + 1\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      2. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right)}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \color{blue}{\left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      4. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \color{blue}{\cos \tan^{-1} x}\right)}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}{1 + \sqrt{\color{blue}{\frac{1}{2} \cdot \left(\cos \tan^{-1} x + 1\right)}}} \]

      6. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}{1 + \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right)}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}{1 + \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

      8. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\cos \tan^{-1} x}\right)}} \]

      9. cos-atan-rev N/A

         \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}}\right)}} \]

      10. sqrt-prod N/A

          \[\leadsto \frac{1 - \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)}{1 + \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{1 + \frac{1}{\sqrt{1 + x \cdot x}}}}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x - -1\right) \cdot 0.5}{\mathsf{fma}\left(\sqrt{\cos \tan^{-1} x - -1}, \sqrt{0.5}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0112:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x\_m - -1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0112)
   (*
    (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
    (* x_m x_m))
   (/
    (- 1.0 (* (- (sqrt (/ 1.0 (fma x_m x_m 1.0))) -1.0) 0.5))
    (+ 1.0 (sqrt (* (- (cos (atan x_m)) -1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0112) {
		tmp = fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - ((sqrt((1.0 / fma(x_m, x_m, 1.0))) - -1.0) * 0.5)) / (1.0 + sqrt(((cos(atan(x_m)) - -1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0112)
		tmp = Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) - -1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(cos(atan(x_m)) - -1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0112], N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0111999999999999999

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2} + \frac{1}{8}\right) \cdot {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      13. lower-*.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0111999999999999999 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. flip-- N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \cdot \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}{1 + \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + x \cdot x}}\right)}}} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   5. Step-by-step derivation

      1. cos-atan-rev N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1 - \left(\frac{\color{blue}{\sqrt{1}}}{\sqrt{1 + x \cdot x}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      3. sqrt-undiv N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\sqrt{\frac{1}{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\sqrt{\frac{1}{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1 - \left(\sqrt{\color{blue}{\frac{1}{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{1 - \left(\sqrt{\frac{1}{1 + \color{blue}{{x}^{2}}}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1 - \left(\sqrt{\frac{1}{\color{blue}{{x}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{1 - \left(\sqrt{\frac{1}{\color{blue}{x \cdot x} + 1}} + 1\right) \cdot \frac{1}{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      9. lower-fma.f64 100.0

         \[\leadsto \frac{1 - \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{1 - \left(\color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0112:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}} - -1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x - -1\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.013:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.013)
   (*
    (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.013) {
		tmp = fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.013)
		tmp = Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.013], N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0129999999999999994

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2} + \frac{1}{8}\right) \cdot {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      13. lower-*.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0129999999999999994 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\sqrt{1 + x \cdot x}}}\right)} \]

      3. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}}\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}}\right)} \]

      5. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{x \cdot x} + 1}}\right)} \]

      6. lower-fma.f64 98.5

         \[\leadsto 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}}\right)} \]

   4. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (*
    (fma
     (-
      (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) (* x_m x_m))
      0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma(((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   7. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{1}{x} + \color{blue}{\frac{1}{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{1}{x} + \color{blue}{\frac{1}{2}}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}} \]

      5. lower-/.f64 98.5

         \[\leadsto 1 - \sqrt{\frac{0.5}{x} + 0.5} \]

   5. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5}{x} + 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.2)
   (*
    (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.2], N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2} + \frac{1}{8}\right) \cdot {x}^{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {x}^{2} \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      13. lower-*.f64 72.6

          \[\leadsto \mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 1.19999999999999996 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{1}{x} + \color{blue}{\frac{1}{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{1}{x} + \color{blue}{\frac{1}{2}}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}} \]

      5. lower-/.f64 98.5

         \[\leadsto 1 - \sqrt{\frac{0.5}{x} + 0.5} \]

   5. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5}{x} + 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma -0.0859375 (* x_m x_m) 0.125) (* x_m x_m))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma(-0.0859375, (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(-0.0859375, Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      8. lower-*.f64 71.7

         \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{1}{x} + \color{blue}{\frac{1}{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{1}{x} + \color{blue}{\frac{1}{2}}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}} \]

      5. lower-/.f64 98.5

         \[\leadsto 1 - \sqrt{\frac{0.5}{x} + 0.5} \]

   5. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5}{x} + 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma -0.0859375 (* x_m x_m) 0.125) (* x_m x_m))
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma(-0.0859375, (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(-0.0859375, Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot {\color{blue}{x}}^{2} \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot {x}^{2} \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-11}{128}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      8. lower-*.f64 71.7

         \[\leadsto \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.8%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.52:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.52) (* (* x_m x_m) 0.125) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.52) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.52d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.52) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.52:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.52)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.52)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.52], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 69.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

      3. frac-add N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      5. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      7. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      8. +-commutative N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      9. pow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      10. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      11. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      12. lower-asinh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

      13. lower-*.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

      14. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

      15. +-commutative N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

      16. pow2 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

      17. cosh-asinh-rev N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      18. lower-cosh.f64 N/A

          \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

      19. lower-asinh.f64 69.8

          \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

   4. Applied rewrites 69.8%

      \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

      3. pow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{8} \]

      4. lower-*.f64 72.3

         \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   7. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.125} \]

   if 1.52 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.8%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot x\_m\right) \cdot 0.125 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (* x_m x_m) 0.125))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * 0.125;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = (x_m * x_m) * 0.125d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * 0.125;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * 0.125

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * 0.125)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) * 0.125;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision]

Derivation

1. Initial program 75.7%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{1} + x \cdot x}}\right)} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \left(\color{blue}{\frac{2}{2}} + \frac{1}{\sqrt{1 + x \cdot x}}\right)} \]

   3. frac-add N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \color{blue}{\frac{2 \cdot \sqrt{1 + x \cdot x} + 2 \cdot 1}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

   5. metadata-eval N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{2 \cdot \sqrt{1 + x \cdot x} + \color{blue}{2}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   6. lower-fma.f64 N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\color{blue}{\mathsf{fma}\left(2, \sqrt{1 + x \cdot x}, 2\right)}}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   7. pow2 N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{1 + \color{blue}{{x}^{2}}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   8. +-commutative N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{{x}^{2} + 1}}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   9. pow2 N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \sqrt{\color{blue}{x \cdot x} + 1}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   10. cosh-asinh-rev N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   11. lower-cosh.f64 N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \color{blue}{\cosh \sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   12. lower-asinh.f64 N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \color{blue}{\sinh^{-1} x}, 2\right)}{2 \cdot \sqrt{1 + x \cdot x}}} \]

   13. lower-*.f64 N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{\color{blue}{2 \cdot \sqrt{1 + x \cdot x}}}} \]

   14. pow2 N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{1 + \color{blue}{{x}^{2}}}}} \]

   15. +-commutative N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{{x}^{2} + 1}}}} \]

   16. pow2 N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \sqrt{\color{blue}{x \cdot x} + 1}}} \]

   17. cosh-asinh-rev N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

   18. lower-cosh.f64 N/A

       \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \color{blue}{\cosh \sinh^{-1} x}}} \]

   19. lower-asinh.f64 75.8

       \[\leadsto 1 - \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \color{blue}{\sinh^{-1} x}}} \]

4. Applied rewrites 75.8%

   \[\leadsto 1 - \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, \cosh \sinh^{-1} x, 2\right)}{2 \cdot \cosh \sinh^{-1} x}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

   2. lower-*.f64 N/A

      \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{8}} \]

   3. pow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{8} \]

   4. lower-*.f64 58.2

      \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

7. Applied rewrites 58.2%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.125} \]
8. Add Preprocessing

Alternative 11: 28.1% accurate, 134.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 75.7%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot 2} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{1} \]

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 33.9

      \[\leadsto 0 \]

5. Applied rewrites 33.9%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025044 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))